Let \(\ell \in \mathcal {S}\)  corresponding to a non-degenerate 2-dimensional
subspace \(U\)  of \(\mathbb {K}^{n}\)  of type \(-\) . Such a
subspace does exists. Indeed, \(U\)  is either \(\left\langle z,\omega z\right\rangle _{\mathbb {K}}\)  or \(\left\langle z,\epsilon z\right\rangle _{\mathbb {K}}\) 
according to whether \(q\)  is equivalent to 1 or 3 modulo 4 by [1]}, Proposition 2.3.(c). Moreover, \(D(Q)=\boxtimes \)  by [2]}, Proposition
2.5.10.(ii), as \(n/2\)  is even. Then \(X_{\ell }\cong \left( \Omega _{2}^{-}(q)\times \Omega _{n-2}^{+}(q)\right) .[4]\)  by [2]}, Proposition 4.1.6.(II). If \(\ell ^{\perp }\)  denotes the \((n-3)\) -dimensional subspace of \(PG_{n-1}(q)\)  corresponding to \(U^{\perp }\) , we see that \(X(\ell ^{\perp })\) , the pointwise stabilizer in \(X\)  of \(\ell ^{\perp }\) , contains \(\Omega _{2}^{-}(q)\) .
The subspace \(U\)  is the image under \(\Phi \)  of a non-degenerate 1-dimensional subspace of \(\mathbb {L}^{n/2}\) . Hence, by [2]}, Proposition 4.1.6.(II), and by [1]}, Proposition 2.3.(c), we see that
\(X_{\mathcal {S},\ell }\cong \left\lbrace \begin{split}& \Omega _{n/2-1}(q^{2})\hphantom{.2}\hphantom{oho} \text{ for } q(n-1)/8 \text{ odd} \\& \Omega _{n/2-1}(q^{2}).2\hphantom{oho} \text{ for } q(n-1)/8 \text{ even.} \\\end{split}\right.\) 
Therefore \( X_{\mathcal {S}}(\ell ^{\perp })=1\) . Thus \( X(\ell ^{\perp }) \ne X_{\mathcal {S}}(\ell ^{\perp })\) , and hence there is \(\alpha \in X(\ell ^{\perp })-X_{\mathcal {S}}(\ell ^{\perp })\)  such that \(\Omega ^{-} _{n/2-1}(q^{2})\leqslant X_{\mathcal {S},\mathcal {S}^{\alpha }}\) .
